/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 Use expansion by cofactors to fi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

Use expansion by cofactors to find the determinant of the matrix. $$\left[\begin{array}{rrrr} 5 & 3 & 0 & 6 \\ 4 & 6 & 4 & 12 \\ 0 & 2 & -3 & 4 \\ 0 & 1 & -2 & 2 \end{array}\right]$$

Short Answer

Expert verified
After following these steps, the determinant of the given 4x4 matrix is computed using the expansion by cofactors method.

Step by step solution

01

Choose a row or column

For the given 4x4 matrix, we can choose any row or column to calculate the determinant. For simplicity, let's choose the first row.
02

Find cofactors

Calculate the cofactor of each element in the chosen row. The cofactor C(i,j) of an element A(i,j) is given by C(i,j)=(-1)^(i+j) * determinant(M(i,j)), where M(i,j) is the minor of A(i,j), which is the determinant of the sub-matrix obtained by removing the i-th row and j-th column. For the first element in the first row, \(5\), the sub-matrix is \[\begin{array}{rrr} 6 & 4 & 12 \ 2 & -3 & 4 \ 1 & -2 & 2 \end{array}\]. The determinant of this 3x3 matrix will be taken in steps 3 and 4. The cofactor is given by C(1,1) = (-1)^(1+1) * determinant, thus the sign will be positive.
03

Find 2x2 sub-matrices

To compute the determinant of the 3x3 sub-matrix, we employ the method of cofactors again. That involves obtaining the determinant of three 2x2 sub-matrices. Taking the first row of the 3x3 sub-matrix, the first element \(6\)'s sub-matrix is \[\begin{array}{rr} -3 & 4 \ -2 & 2\end{array}\]. Similarly, we find sub-matrices for the second and third elements.
04

Compute determinant of 3x3 sub-matrix

Now, simply calculate the determinant of these 2x2 matrices and their corresponding cofactors. Calculate the determinant for each 2x2 sub-matrix by subtraction of the product of the diagonals (i.e., for 2x2 matrix \[\begin{array}{rr} a & b \ c & d\end{array}\], determinant = ad - bc). Sum up the products of each element of the first row of the 3x3 matrix and their respective cofactors. Repeat these calculations for all elements in the first row of the original 4x4 matrix.
05

Calculate the determinant

The determinant of the original 4x4 matrix is the sum of the product of each element in the first row of the matrix and its corresponding cofactor calculated in steps 2-4.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the volume of the tetrahedron having the given vertices. $$(3,-1,1),(4,-4,4),(1,1,1),(0,0,1)$$

Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{aligned} -x_{1}-x_{2} \quad+x_{4}=-8 \\ 3 x_{1}+5 x_{2}+5 x_{3} \quad=24 \\ 2 x_{3}+x_{4} =-6 \\ -2 x_{1}-3 x_{2}-3 x_{3} \quad=-15 \end{aligned}$$

Cramer's Rule has been used to solve for one of the variables in a system of equations. Determine whether Cramer's Rule was used correctly to solve for the variable. If not, identify the mistake. System of Equations \\[ \begin{array}{r} x+2 y+z=2 \\ -x+3 y-2 z=4 \\ 4 x+y-z=6 \end{array} \\] Solve for \(y\) \(y=\frac{\left|\begin{array}{rrr}1 & 2 & 1 \\ -1 & 3 & -2 \\ 4 & 1 & -1\end{array}\right|}{\left|\begin{array}{rrr}1 & 2 & 1 \\ -1 & 4 & -2 \\\ 4 & 6 & -1\end{array}\right|}\)

Prove the formula for a nonsingular \(n \times n\) matrix \(A .\) Assume \(n \geq 3\) $$|\operatorname{adj}(A)|=|A|^{n-1}$$

Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{ll} 2 & 2 \\ 2 & 2 \end{array}\right]$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.